Some Problems in Stochastic Flows and Diffusions

点击翻译按钮获取中文摘要

This research encompasses problems in the two areas of stochastic flows and coupling of diffusions. The principal problems in the latter area involve a probabilistic proof of Moser's Harnack inequality and a search for the best coupling on manifolds. In the former area, the problems concern the qualitative and quantitative behavior of Brownian and turbulent flows, especially their dispersion properties.The general setting of the set of problems on coupling is the behavior of steady state temperature distributions on manifolds (surfaces for example) and the rate of convergence to this steady state. The scope of the work on stochastic flows is quite broad. It presently concerns the nature of dispersion of oil slicks or high temperature bodies on the ocean's surface. The ultimate goal is to apply some of the ideas from previous and current work to solve problems about the fractal nature of the energy of the magnetic field on the sun.